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RT&A 2017, # 4(47) Vol.12


 

Yastrebenetsky M.

 

In anniversary of professor Alexander Solov’ev

 

Professor Alexander Dmitrievich Solov’ev (1927-2001), professor of Lomonosov Moscow State University, was one of the founders of reliability theory, author the classical books, the nearest colleague of academic Boris Gnedenko. Aim of this short article- to mark   90-th birth anniversary of Alexander Solov’ev.  

 

DOI: https://doi.org/10.24411/1932-2321-2017-14001

Key words: reliability theory, mathematical methods, Solov’ev

 

 

Krishnamoorthy A., Vishnevsky V., Manjunath A. S., Dhanya Shajin

 

Single server queues with several services

 

A service station offers several services of which one is the required and the others are harmful (or destructive) for each customer. At the time when selected for service customers enter in correct mode of service according to a Bernoulli process with parameter  which is the probability of being selected in correct mode. Arrival follows Markovian arrival process and service time is Phase type distributed in both undesired and required phases. An exponentially distributed threshold clock starts ticking if a customer enters to incorrect mode and the service is terminated if the clock realizes before the customer is transferred to correct service mode. The rate of loss of customers, rate of customers leaving with correct service starting with incorrect service are computed. Authors examined a queueing model offering  distinct services to which arrival is according to a  forming a single line. Service time has phase type distribution. A single server serves the customers. The service station provides two types of services - one is desirable and other is unwanted for each customer. If the service starts in an undesirable state then a clock also simultaneously starts ticking. In case this clock realizes before the exact requirement of the server is realized, then that customer leaves the system forever without being eligible for the service that he actually requires. On the other extreme, in case the correct identification of required service occurs before realization of clock, then the customer is served in that state and then leaves the system. In case right at the beginning of service the exact requirement of service is identified, then the customer starts getting that service right at the time when taken for service. Several system performance measures are evaluated. Applications of the model in hospital, telecommunication etc are indicated. Stochastic decomposition of the system state is analyzed. Some particular cases are indicated.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14002

Keywords: desired/undesired service states, random threshold clock, Markovian arrival process

 

 

Anulova S.V., Mai H., Veretennikov A.Yu.

 

On averaged expected cost control as reliability for 1D ergodic diffusions
 

For a Markov model described by a one-dimensional diffusion with ergodic control without discount on the infinite horizon an ergodic Bellman equation is proved for the optimal readiness coefficient; convergence of the iteration improvement algorithm is established.

DOI: https://doi.org/10.24411/1932-2321-2017-14003  

 

 

Abramov O.V., Dimitrov B.

 

Reliability Design In Gradual Failures: Functional-Parametrical Approach

 

This conceptual paper discusses the main provisions of the functional-parametrical (FP) approach in reliability studies. The FP itself is a detailed frame algorithm suggested for use in design of new and unique technical items. The article also presents possibilities and perspectives of using FP approach in problems for “building in” reliability in analogy to these for technical devices and systems. It is pointed out that for solving problems of analysis and ensuring of desired reliability it is appropriate to use parallel and distributed processing techniques. We discuss the idea of constructing efficient parallel algorithms for multivariate statistical analysis necessary to calculate estimates of the probability for failure-free operation with different nominal values of internal parameters. More use of parallel algorithms, including in continuous media via discretization are discussed.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14004 

Keywords: computational methods, conceptual algorithms, gradual failure, parallel computing, parametric synthesis, projected reliability

 

 

Sorokovikova O., Dzama D., Asfandiyarov D., Blagodatskikh D.

 

Probabilistic Dispersion Models In Large Water Areas. Statistics And Stochastic Computational Algorythms

 

Probabilistic Dispersion Models In Large Water Areas. Statistics And Stochastic Computational Algorythms

A Monte Carlo method for calculation of dispersion in large water areas with complex coastlines is presented. Having utilized a multi-year database of sea currents and of distributions of the mixed layer depth, a model of statistical ranking of water areas based on contamination levels is proposed. Techniques for parallelization of the Monte-Carlo method and statistical postprocessing of results have been realized. The method and the techniques have been validated against releases into a particular large water area. In this article a Lagrangian stochastic model for calculation of ocean dispersion and probabilistic approaches to estimations of simulation results are presented. It was demonstrated in the case of a hypothetical emergency situation at Razboynik Bay that stable distribution patterns of contamination may occur in a large water area. Therefore, it is confirmed that an analysis utilizing the methods described in this article can be used for optimization of sampling strategy near coastal nuclear legacy repositories under assumption of the seasonal variability of ocean currents.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14005

Keywords: Monte Carlo method, ocean dispersion model, pseudorandom number generator, parallel calculations

 

Rusev V., Skorikov A.

 

On Solution of Renewal Equation in the Weibull-Gnedenko Model

 

Renewal density of restorable systems and their components which depends on statistical estimates based on real operational data is studied. It is assumed that objects’ entire life cycle is described by the Weibull-Gnedenko distribution. Analytical and discrete approaches for the solution to the renewal equation are proposed. New calculation schemes of the renewal density of restorable systems and their components are presented. Equivalence of suggested approaches is illustrated by numerical examples.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14006

Keywords: the Weibull-Gnedenko distribution, reliability theory, renewal density (intensity), numerical methods, collocation knots, moments generating function

 

 

Ivnitskiy V.

 

Flow Thinning With Limited Aftereffect: Differently Distributed Intervals Between the Moments of Customers Entrance

 

We present some analytical results obtained for probability characteristics of  flow thinning  with limited aftereffect. The thinning is processed according to a given function which depends on the evolution time and on the number  of customers in the thinned flow and the  number of lost customers in the original flow. The characteristics are obtained in the form of Laplace-Stieltjes transforms which are defined by the system of recurrence equations  using the  inverse  Laplace-Stieltjes transform. Thus, in the article, obtained some analytical results for probability characteristics of a thinning of the flow with different-distributed intervals between the moments of customers entrance (flow with limited aftereffect). The thinning is processed according to a given function which depends on evolution time and on the number customers of the thinned flow and the number of lost customers in theoriginal flow. The characteristics are obtained in the form of Laplace-Stieltjes transforms which are defined by the system of recurrence equations with using inversion of Laplace-Stieltjes transforms.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14007

Keywords: flow with limited aftereffect, thinning of the flow, time-dependent function of thinning, Laplace-Stieltjes transform, inverse Laplace-Stieltjes transform.

 

 

Kamal Ullah, Inthekhab Alam, Showkat Ahmad Lone

 

Accelerated Life Testing Design Using Geometric Process for Generalized Rayleigh Distribution with Complete Data

 

The log-linear function between life and stress which is just a simple re-parameterization of the original parameter of the life distribution is used to obtain the estimates of original parameters in many of the studies concerning Accelerated life testing (ALT). But from the statistical point of view, it is preferable to work with the original parameters instead of developing inferences for the parameters of the log-linear link function.  In this study we introduce the geometric process for the analysis of accelerated life testing with Generalized Rayleigh Distribution for constant stress. Assuming that the lifetimes of units under increasing stress levels form a geometric process, the maximum likelihood estimation approach is used for the estimation of parameters. The confidence intervals (CIs) of the model parameters are derived. A Simulation study is also performed to check the statistical properties of estimates of the parameters and the confidence intervals. In this study, the geometric process is introduced for the analysis of accelerated life testing under constant stress when the life data are from a generalized Rayleigh distribution. It is a better choice for life testing because of its simplicity in nature. The Mean, SE and RMSE of the parameters are obtained. Based on the asymptotic normality, the 95% and 99% confidence intervals of the parameters are also obtained. The results show in article that the estimated values of  β and λ are very close to true (or initial) value with very small SE and RMSE. As sample size increases, the value of SE and RMSE decreases and the confidence interval become narrower. 

 

DOI: https://doi.org/10.24411/1932-2321-2017-14008

Keywords: Geometric process, Generalized Rayleigh Distribution, Maximum Likelihood Estimator, Fisher Information Matrix, Asymptotic Confidence Interval, Simulation Study.

 

 

Rakesh Kumar, Sapana Sharma

 

Time-Dependent Analysis of a Single-Server Queuing Model with Discouraged Arrivals and Retention of Reneging Customers

 

In this paper, a finite capacity Markovian single-server queuing system with discouraged arrivals, reneging, and retention of reneging customers is studied. The time-dependent probabilities of the queuing system are obtained by using a computational technique based on the 4th order Runge-Kutta method. With the help of the time-dependent probabilities, we develop some important measures of performance of the system, such as expected system size, expected reneging rate, and expected retention rate. The time-dependent behavior of the system size probabilities and the expected system size is also studied. Further, the variations in the expected system size, the expected reneging rate, and the expected retention rate with respect to the probability of retaining a reneging customer are also studied. Finally, the effect of discouragement in the same model is analyzed. The time-dependent analysis of a single-server queuing system with discouraged arrivals, reneging and retention of reneging customers is performed by using Runge-Kutta method. The numerical results are computed with the help of MATLAB software. The effect of probability of retaining a reneging customer on various performance measures is studied. We also study the impact of discouraged arrivals on the system performance.

 

DOI: https://doi.org/10.24411/1932-2321-2017-14009

Keywords: time-dependent analysis, single server queuing system, discouraged arrivals, reneging, Runge-Kutta method, retention.

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